Iteration processes with errors for nonlinear equations involving \(\alpha\)-strongly accretive operators in Banach spaces. (Q2785287)

Let \(X\) be a real Banach space and let \(A:X\longrightarrow 2^X\) be an \(\alpha\)-strongly accretive operator. Assume that the duality mapping \(J\) satisfies Calvert and Gupta's condition: NEWLINE\[NEWLINE \sup\{\| f-g\| : f\in J(u), \;g\in J(v)\} \leq \Phi (u-v)NEWLINE\]NEWLINE for all \(u,v\in X\), where \(\Phi : X\rightarrow [0,\infty )\) is a given function. The author shows that this condition together with some additional assumptions implies the strong convergence of the so-called Ishikawa and Mann iteration processes (associated with \(A\)) to the unique solution of the operator equation \(z\in Ax\). In addition, the convergence of the Ishikawa and Mann iteration processes with errors for \(\alpha\)-strongly pseudo-contractive operators is proved.